Abstract

We study random, finite-dimensional, ungraded chain complexes over a finite
field and show that for a uniformly distributed differential a complex has the
smallest possible homology with the highest probability: either zero or one-
dimensional homology depending on the parity of the dimension of the complex.
We prove that as the order of the field goes to infinity the probability distribution
concentrates in the smallest possible dimension of the homology. On the other
hand, the limit probability distribution, as the dimension of the complex goes
to infinity, is a super-exponentially decreasing, but strictly positive, function of
the dimension of the homology.

Keywords

Random chain
complexes, Homology, Floer theory

Mathematics Subject Classification

05E99, 55U15, 53D99, 60D99

1. Introduction

We study random, finite-dimensional, ungraded chain complexes over
a finite field and we are interested in the probability that such a complex has
homology of a given dimension. We show that for a uniformly distributed
differential the complex has the smallest possible homology with the highest
probability.

To be more specific, consider an $ n$-dimensional vector space
$ V$ over a finite field $ \F=\F_q$ of order $ q$ and let
$ D$ be a differential on $ V$, i.e., a linear operator $ D\colon V\to V$
with $ D^2=0$. We are interested in the probability $ p_r(q,n)$ with
which a chain complex $ (V,D)$ has homology $ \ker D/\im D$ of a given
dimension $ r$ for fixed $ n$ and $ q$. The
differential $ D$ is uniformly distributed and $ p_r(q,n)$ is simply
the ratio $ c_r(q,n)/c(q,n)$, where $ c_r(q,n)$ is the number of complexes with
$ r$-dimensional homology and $ c(q,n)$ is the number of all
complexes. The numbers $ c_r(q,n)$ are explicitly calculated by [Kovacs1987]; see also Theorem 2.2.

We mainly focus on large complexes, i.e., on the limits as
$ q$ or $ n$ go to infinity. Clearly, $ r$ and
$ n$ must have the same parity and we separately analyze the
asymptotic behavior of the sequence $ p_0(q,n), p_2(q,n),\ldots $, where $ n$ is even,
and the sequence $ p_1(q,n), p_3(q,n),\ldots $ for $ n$ odd.

As $ q\to\infty$ with $ n$ fixed, the probability concentrates
in the lowest possible dimension, i.e., $ p_0(q,n)\to 1$ or $ p_1(q,n)\to 1$ depending
on the parity of $ n$, while $ p_r(q,n)\to 0$ for $ r>1$. This is
consistent with the observation that over $ \C$ and even $ \R$
(see Lemma 3.1) a generic complex has $ 0$- or
$ 1$-dimensional homology, i.e., that such complexes form the highest
dimensional stratum in the variety of all $ n$-dimensional complexes.
Indeed, one can expect the probability distributions for large $ q$ to
approximate the generic situation in zero characteristic. We do not know,
however, if the density functions converge in any sense as $ q\to\infty$ to
some probability density on the variety of $ n$-dimensional complexes
over, e.g., $ \R$.

When $ q$ and $ r$ are fixed and $ n\to\infty$
through either even or odd integers depending on the parity of $ r$,
the situation is more subtle. In this case, all limit probabilities $ p_r(q)=\lim_{n\to\infty}p_r(q,n)$
are positive. However, the sequences $ p_0,p_2,\ldots$ and $ p_1,p_3,\ldots$ are
super-exponentially decreasing and for a large $ q$ all terms in these
sequences but the first one are very close to zero while the first is then, of
course, close to 1. When $ q=2$ and $ r$ is even, we have
$ p_0\approx 0.6$, $ p_2\approx 0.4$, $ p_4\approx 0.0075$ and other terms are very small. We
explicitly calculate the ratios $ p_r(q)/p_0(q)$ and $ p_r(q)/p_1(q)$ and $ p_0$
and $ p_1$ in Theorem 2.1.

The proofs of these facts are elementary and quite simple. However,
we have not been able to find in the literature any probability calculations in
this basic case where random chain complexes are stripped of all additional
structures including a grading. (The combinatorial part of our proof, Theorem
2.2, is
contained in [[Kovacs1987], Lemma 5].) In contrast, random
complexes of geometrical origin and underlying random geometrical and
topological objects have been studied extensively and from various
perspectives. Among such random objects are, for instance, random simplicial
complexes of various types [see ([Aronshtam et al.2013],[Bobrowski and Kahle2014],[Costa and Farber2014],[Costa and Farber2015],[Kahle2011],[Meshulam2013],[Meshulam and Wallach2009],[Pippenger and Schleich2006],[Yogeshwaran et al.2014]) and references therein]
and random Morse functions [see, e.g., ([Arnold2006],[Arnold2007],[Collier et al.2017],[Nicolaescu2012])].

These works utilize several models of randomness all of which appear
to be quite different from the one, admittedly rather naive, used here. This
makes a direct comparison difficult. One way to interpret our result is that, for
a large complex, sufficiently non-trivial homology is indicative of some structure,
a constraint limiting randomness. Note that such a structure can be as simple
as a $ \Z$-grading confined to a fixed range of degrees. A dimensional
constraint of this type is usually inherent in geometrical complexes, and it would
be interesting to analyze its effect (if any) on the probability distribution in our
purely algebraic setting. Another consequence of the result is that the assertion
that a complex has large homology carries more information than the assertion
that it has small homology.

The main motivation for our setting comes from Hamiltonian Floer
theory for closed symplectic manifolds; see, e.g., [Salamon1999] and references therein. A
Hamiltonian diffeomorphism is the time-one map of the isotopy generated by a
time-dependent Hamiltonian. To such a diffeomorphism one can associate a
certain complex, called the Floer complex, generated by its fixed points or,
equivalently, the one-periodic orbits of the isotopy. Hence the dimension of the
Floer homology gives a lower bound for the number of one-periodic orbits.
The homology is independent of the Hamiltonian diffeomorphism. In addition,
one can fix the free homotopy class of the orbits. (This construction is similar
to Morse theory and, in fact, Floer theory is a version of Morse theory for the
action functional.)

In many instances, e.g., often generically or for all symplectic
manifolds with vanishing first Chern class such as tori, the dimension of the
Floer complex grows with the order of iteration of the diffeomorphism; see
[Ginzburg and G[U+00A8]urel2015]. In other words, the
complex gets larger and larger as time in this discrete dynamical system grows.
Moreover, the differential in the complex is usually impossible to describe
explicitly, and hence it makes sense to compare the behavior of the complex
and its homology with the generic or random situation. The Floer homology
for contractible periodic orbits is isomorphic to the homology of the underlying
manifold. Therefore, by our result, even though the Floer complex appears to
be very "noisy" for large iterations and random on a bounded action scale, it
has large homology groups and is actually very far from random. For
non-contractible orbits, the dimension of the Floer complex is also known to
grow in many settings; see [Ginzburg and G[U+00A8]urel2015],[G[U+00A8]urel2013]. However, in this case the Floer
homology is zero and the complex may well be close to random. Note also that
in some instances the Floer complex is $ \Z$-graded, but the grading is
not supported within any specific interval of degrees. Moreover, in contrast
with geometrical random complexes, the grading range of the Floer homology
usually grows with the order of iteration ([Salamon and Zehnder1992]), and while it is not
clear how to correctly account for an unbounded grading in a random model,
such a grading is unlikely to affect the probability distribution.

One aspect of Floer theory which is completely ignored in our model
is the action filtration. This filtration is extremely important and, in
particular, it allows one to treat Floer theory in the context of persistent
homology and topological data analysis; see [Carlsson2009],[Ghrist2008]. This connection has recently been
explored in [Polterovich and Shelukhin2014],[Usher and Zhang2015]. However, it is not entirely
clear how to meaningfully incorporate the action filtration into our model.

2. Main Results

Let, as in the introduction, $ (V,D)$ be an ungraded
$ n$-dimensional chain complex with differential $ D$ over a
finite field $ \F=\F_q$ of order $ q$. In other words, $ V=\F^n$
and $ D$ is a linear operator on $ V$ with $ D^2=0$. We
denote by $ c(q,n)$ the number of such complexes, i.e., the number of
differentials $ D$. The dimension $ r$ of the homology
$ \ker D/\im D$ has the same parity as $ n$ and we let $ c_r(q,n)$ be
the number of complexes with homology of dimension $ r$. (In what
follows, we always assume that $ r$ and $ n$ have the same
parity.) Clearly, \begin{eqnarray*} c(q,n)=c_0(q,n)+c_2(q,n)+\cdots+c_{n}(q,n) \end{eqnarray*} when $ n$ is even and \begin{eqnarray*} c(q,n)=c_1(q,n)+c_3(q,n)+\cdots+c_{n}(q,n) \end{eqnarray*} when
$ n$ is odd.

Furthermore, denote by \begin{eqnarray*} p_r(q,n)=\frac{c_r(q,n)}{c(q,n)} \end{eqnarray*} the probability (with respect to
the uniform distribution) of a complex to have $ r$-dimensional
homology. Our main result describes the behavior of $ p_r(q,n)$ as the size
of the complex, i.e., $ q$ or $ n$, goes to infinity.

Theorem 2.1.

Let $ p_r(q,n)$ be as above.

(i) For a fixed
$ n$, we have \begin{equation*} \lim_{q\to\infty} p_r(q,n)=0\textrm{ when } r>1, \end{equation*} and $ p_0(q,n)\to 1$ when $ n$ is even
and $ p_1(q,n)\to 1$ when $ n$ is odd as $ q\to\infty$.

The proof of this theorem is based on an explicit calculation of
$ c_r(q,n)$. To state the result, denote by $ \GL_k(q)$ the general linear
group of $ k\times k$ invertible matrices over $ \F_q$ and recall that
\begin{eqnarray*} |\GL_k(q)|=q^{k(k-1)/2}\prod_{j=1}^k(q^j-1). \end{eqnarray*} Then we have the following particular case of [[Kovacs1987], Lemma 5].

Theorem 2.2.

([Kovacs1987]) Let as above $ c_r(q,n)$ be the
number of $ n$-dimensional complexes over $ \F_q$ with
homology of dimension $ r$. Then

Even though this
result is not new, for the sake of completeness we include its proof, which is
very simple and short, in the next section.

Remark 2.3.

We do not have simple expressions for the
probabilities $ p_r(q,n)$ and the total number of complexes $ c(q,n)$.
However, when $ q=2$, the differentials $ D$ are in one-to-one
correspondence with involutions of $ \F_2^n$. (An involution necessarily
has the form $ I+D$ and, as is easy to see, different differentials
$ D$ give rise to different involutions.) Hence, $ c(2,n)$ is equal
to the number of involutions. This number is expressed in [Fulman and Vinroot2014] via a generating function
and an asymptotic formula for $ c(q,n)$ has been recently obtained in
[Fulman et al.2016]. It is possible that at least
when $ q=2$ our probability formulas can be further simplified using
the results from those papers.

3. Proofs

The proof of Theorem 2.2 is based on the observation that the differential
in a finite-dimensional complex over any field $ \F$ can be brought to
its Jordan normal form or, equivalently, a complex over $ \F$ can be
decomposed into a sum of elementary complexes, i.e., into a sum of
two-dimensional complexes with zero homology and one-dimensional
complexes. To be more precise, we have the following elementary observation.

Lemma 3.1.

Let $ V$ be a finite-dimensional vector
space over an arbitrary field $ \F$ and let $ D\colon V\to V$ be an
operator with $ D^2=0$. Then, in some basis, $ D$ can be
written as a direct sum of $ 1\times 1$ and $ 2\times 2$ Jordan blocks with
zero eigenvalues.

When $ \F$ is algebraically closed, this follows immediately
from the Jordan normal form theorem. Hence, the emphasis here is on the fact
that the field $ \F$ is immaterial. For the sake of completeness, we
outline a proof of the lemma.

Proof.

Proof of Theorem 2.2
Let $ D$ be a differential on an $ n$-dimensional vector
space $ V$ over a finite field $ \F=\F_q$. Assume that the
homology of the complex $ (V,D)$ is $ r$-dimensional. By
Lemma 3.1, $ D$ is conjugate to the map
$ D_r$ which is the direct sum of $ r$ $ 1\times 1$ zero
blocks and $ m$ $ 2\times 2$ Jordan blocks with zero eigenvalues,
where $ 2m+r=n$.

Let $ C_r$ be the centralizer of $ D_r$ in
$ \GL_n(q)$. The complexes with $ r$-dimensional homology are
in one-to-one correspondence with $ \GL_n(q)/C_r$. Thus, to prove
(2.3)
, it suffices to show that

The elements of $ C_r$ are $ n\times n$ invertible matrices
$ X\in \GL_n(q)$ commuting with $ D_r$. In what follows, it is convenient
to work with the basis $ e_1, \ldots, e_m, f_1, \ldots, f_r, e'_1,\ldots,e'_m$ in the notation from the proof of Lemma
3.1. Thus
we can think of $ X$ as a $ 3\times 3$-block matrix with
$ m\times m$ block $ X_{11}$, the block $ X_{12}$ having size
$ m\times r$, and $ X_{13}$ being again $ m\times m$, etc. In the same
format, $ D_r$ is then the matrix with only one non-zero block. This is
the top-right corner $ m\times m$-block, which is $ I$. Then, as a
straightforward calculation shows, the commutation relation $ XD_r=D_rX$
amounts to the conditions that $ X_{11}=X_{33}$, and $ X_{21}=0$,
$ X_{31}=0$ and $ X_{32}=0$. In particular, $ X$ is an upper
block-triangular matrix. Hence, $ X$ is invertible if and only if
$ X_{11}=X_{33}$ and $ X_{22}$ are invertible. There are no constraints on
the remaining blocks $ X_{12}$, $ X_{13}$ and $ X_{23}$. Now
(3.1)
follows. ⬜

Proof of Theorem
2.1.

Throughout the proof, we assume that
$ r$ and $ n$ are even. The case where these parameters
are odd can be handled in a similar fashion.

As the first step, we express $ p_r(q,n)/p_0(q,n)$ explicitly. Clearly,
\begin{eqnarray*} \frac{p_r(q,n)}{p_0(q,n)}=\frac{c_r(q,n)}{c_0(q,n)}=\frac{|C_0|}{|C_r|}. \end{eqnarray*} Using
(2.3)
or
(3.1)
and tidying up the resulting expression, we have \begin{align*} \frac{p_r(q,n)}{p_0(q,n)} &=\frac{q^{n^2/4}\cdot q^{n(n/2-1)/4}\cdot \prod_{j=1}^{n/2}(q^j-1)} {q^{2mr+m^2}\cdot q^{m(m-1)/2}\cdot q^{r(r-1)/2}\cdot\prod\nolimits_{j=1}^{r}(q^j-1) \cdot\prod\nolimits_{j=1}^{m}(q^j-1)} \\ &=\frac{\prod\nolimits_{j=m+1}^{m+r/2}(q^j-1)} {q^{mr/2+r(r/2-1)/4}\cdot \prod\nolimits_{j=1}^{r}(q^j-1) }, \end{align*} where as above
$ n=2m+r$ and $ r\geq 2$, which we can then rewrite as

Now it is clear that \begin{eqnarray*} \frac{p_r(q,n)}{p_0(q,n)}\sim q^{-r^2/2}\quad\textrm{ as }\quad q\to\infty \end{eqnarray*} with $ r\geq 2$ and $ n$
fixed. In particular, this ratio goes to zero as $ q\to \infty$. The number of
the terms in the sum \begin{eqnarray*} \sum_j p_j(q,n)=1 \end{eqnarray*} with $ j$ ranging through even
integers from $ 0$ to $ n$ is equal to $ n/2+1$ and
thus this number is independent of $ q$. Hence, $ p_0(q,n)\to 1$ and
$ p_r(q,n)\to 0$ when $ r\geq 2$ as $ q\to\infty$. This proves the first
assertion of the theorem.

Furthermore, in a similar vein, it is not hard to show that \begin{eqnarray*} \sum_{r>0} \frac{p_r(q,n)}{p_0(q,n)}\to S:=\sum_r \frac{q^{r/2}}{\prod\nolimits_{j=1}^r(q^j-1)}\quad\textrm{ as } n\to\infty, \end{eqnarray*}
where, on the left, the sum is taken over all even integers from $ 2$
to $ n$ and, on the right, the sum is over all even integers
$ r\geq 2$. Therefore, letting $ n\to\infty$ in the identity \begin{eqnarray*} 1+\sum_{r>0} \frac{p_r(q,n)}{p_0(q,n)}=\frac{1}{p_0(q,n)}, \end{eqnarray*} we
conclude that the limit $ p_0(q)=\lim_{n\to\infty}p_0(q,n)$ exists and $ p_0(q)=1/(1+S)$, which proves
(2.2)
. Now, by
(3.3)
, the limits $ p_r(q)=\lim_{n\to\infty}p_r(q,n)$ for $ r\geq 2$ also exist, and hence
(2.1)
holds. This completes the proof of the theorem. ⬜

Remark 3.2.

The sequence $ p_r(q,n)$ is decreasing as a
function of $ r$. This readily follows from
(3.2)
.

Acknowledgements.

The authors are grateful to
Robert Ghrist, Ba[U+00B8]sak G[U+00A8]urel, Jiang-Hua Lu, Roy Meshulam and Leonid
Polterovich for useful discussions and comments. The authors would also like
to thank the referee for pointing out [[Kovacs1987], Lemma 5] to them. A part of this
work was carried out while the second author was visiting the Simons Institute
for the Theory of Computing and he would like to thank the institute for its
warm hospitality.